Edit Images Externally With Linux Mint
Typically, you should be able to leave User Preferences > File > Image Editor blank - as Blender will attempt to find 'gimp' on the system path. The 'Image Editor' preference is there as a means of selecting an alternative Image Editor or specifying where to find it if it isn't already on the path.

Note also that the '.desktop' file is a means of your window manager opening an application - Blender should not be using that as it will invoke 'gimp' directly. However, if you open the '.desktop' file then it will tell you what command clicking the 'Gimp' desktop icon will invoke (ie, where Gimp is installed if it isn't on your path).

Firstly, ensure that 'gimp' is on the path. You can do this by opening a 'terminal' window (I'm not too familiar with Mint but it should be something like 'Terminal', 'XTerm', 'Term', 'Command Window' or something similar) to get a command prompt. Then enter 'type gimp'. This should come back with something like :

gimp is hashed (/usr/bin/gimp)


This tells you that Gimp is available in /usr/bin/gimp and is on the system path.

If this is the case then you should simply be able to leave the 'Image Editor' in the File preferences blank - Blender will find it.

If gimp cannot be found (the 'type' command didn't locate it) then you'll need to specify the full path to 'gimp' (not the '.desktop' file). Locate it on your system and set the 'Image Editor' to that full path - so if 'gimp' is located at /usr/local/myversionofgimp/gimp then set the Image Editor to the full path to your 'gimp' executable (ie, '/usr/local/myversionofgimp/gimp').